Statistics 402C
March 29, 2000
Exam 2
Name:
INSTRUCTIONS: Read the questions carefully and completely. Answer the questions and show
work in the space provided. This is the only work that I will look at. Partial credit cannot be
given if work is not shown. Refer to the computer printout and graphs provided when appropriate.
Pace yourself, do not spend too much time on any one problem. Point values for each problem
are given.
1. [15 pts] For the following situations give the name of the design used to collect the data. Give
a partial analysis of variance table indicating all sources of variability and their associated
degrees of freedom.
(a) [5] A mechanical engineer is studying the thrust force developed by a drill press. She
suspects that feed rate of the material is the most important factor. She has four feed
rates. There are 10 operators who normally use the drill press. Each operator uses the
drill press at each of the four feed rates. The order of the feed rates is randomized for
each operator.
(b) [5] An experiment is performed to see whether different operators obtain different results
in the routine analysis to determine the amount of nitrogen in soil. 50 soil samples are
chosen at random and divided at random into 5 groups of 10. Each operator is assigned
a group of 10 soil samples at random and asked to determine the amount of nitrogen in
each sample.
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(c) [5] Similar to the experiment in (a) the thrust force of drill presses is of interest. This
time the engineer is interested in both drill speed and feed rate. The engineer has 10
drill presses, chosen at random, to work with. Five of the drill presses are assigned at
random to run at the high speed. The other five are run at the low speed. For each drill
press each of the 4 feed rates is run in a random order.
2. [25 pts] An experiment is performed to evaluate the effect of various gasoline additives on
the amount of nitrogen oxides in automobile emissions. Since different cars may naturally
produce different amounts and drivers may affect the amount of nitrogen oxides in the emissions, both car and driver are treated as nuisance variables. In the experiment, each driver is
to drive each car using one of the additives. The experiment is designed as a Latin Square.
Below are the data. SAS output is also included.
I
D
R
I
V
E
R
II
III
IV
Mean
A
D
B
C
1
B
21
23
15
2
Car
D
26
C
26
D
13
A
A
C
B
3
C
20
20
16
B
A
D
4
Mean
25
27
16
17
15
20
20
19
20
19
22
2
23
24
15
18
20
Class
DRIVER
CAR
ADDITIVE
Levels
4
4
4
Values
I II III IV
1 2 3 4
A B C D
Number of observations in data set = 16
Dependent Variable: OXIDE
Source
DF
Sum of
Squares
Mean
Square
F Value
Pr > F
Model
9
280.000000
31.111111
11.67
0.0037
Error
6
16.000000
2.666667
15
296.000000
R-Square
0.945946
C.V.
8.164966
Root MSE
1.63299
DF
Type I SS
Mean Square
F Value
Pr > F
3
3
3
216.000000
24.000000
40.000000
72.000000
8.000000
13.333333
27.00
3.00
5.00
0.0007
0.1170
0.0452
Corrected Total
Source
DRIVER
CAR
ADDITIVE
OXIDE Mean
20.0000
T tests (LSD) for variable: OXIDE
NOTE: This test controls the type I comparisonwise error rate not
the experimentwise error rate.
Alpha= 0.05 df= 6 MSE= 2.666667
Critical Value of T= 2.45
Least Significant Difference= 2.8255
Means with the same letter are not significantly different.
T Grouping
Mean
N
ADDITIVE
A
A
A
22.000
4
B
21.000
4
C
19.000
4
D
18.000
4
A
B
B
B
C
C
C
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(a) [5] Are there statistically significant differences among the four additives in terms of
average nitrogen oxide levels? Report the appropriate F value and P-value, and explain
how these support your answer.
(b) [5] What is the value of the Least Significant Difference (LSD)? Using this LSD is there
a significant difference between additive A and additive B?
(c) [15] Although this was designed as a Latin Square, there was some confusion on the part
of the motor pool supplying the cars. Instead of using only 4 cars, 16 cars were used
and these were completely randomized. Therefore, the only nuisance factor accounted
for in the design is the driver. Construct an appropriate ANOVA table for this design
and analysis. Answer the questions in (a) and (b) for this design and analysis.
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3. [30 pts] In order to increase the strength, refine the grain and homogenize the structure of
steel, steel is heated above a critical temperature, soaked and then air cooled. An experiment
is performed to determine the effect of temperature and heat treatment time on the strength
of steel. Two temperatures and three times are selected. The experiment is performed by
heating the oven to a randomly selected temperature and inserting 3 specimens. After 10
minutes one specimen is chosen at random and removed, after 20 minutes a second specimen
is chosen at random and removed, and after 30 minutes the final specimen is removed. Then
the temperature is reset and the process is repeated. Each temperature is replicated 4 times
in a completely random order. Below are the data arising from this split plot design and
output from SAS. Run gives the randomized order of the Temperature settings.
1500 F
Temperature
1600 F
Time
10 20 30 Run
Time
10 20 30
3
52 54 61
7
89 91
62
8
50 52 59
2
72 80
69
1
58 64 71
4
73 81
69
6
48 54 59
5
88 92
64
Run
(a) [2] What is the whole plot factor?
(b) [2] What is the sub plot factor?
(c) [6] Are the two temperatures significantly different in terms of mean strength? Report
the appropriate F- and P- values to support your answer.
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Class Level Information
Class
Levels
Values
TEMP
2
1500 1600
RUN
8
1 2 3 4 5 6 7 8
TIME
3
10 20 30
Number of observations in data set = 24
Dependent Variable: STRENGTH
Source
DF
Sum of
Squares
Model
11
4022.66667
365.69697
Error
12
278.66667
23.22222
Corrected Total
23
4301.33333
R-Square
C.V.
Root MSE
STRENGTH Mean
0.935214
7.174607
4.81894
67.1667
DF
Anova SS
Mean Square
F Value
Pr > F
1
6
2
2
2562.66667
381.33333
192.33333
886.33333
2562.66667
63.55556
96.16667
443.16667
110.35
2.74
4.14
19.08
0.0001
0.0650
0.0429
0.0002
Source
TEMP
RUN(TEMP)
TIME
TEMP*TIME
Level of
TEMP
1500
1600
N
N
10
20
30
8
8
8
F Value
Pr > F
15.75
0.0001
-----------STRENGTH---------Mean
SD
12
12
Level of
TIME
Mean
Square
56.8333333
77.5000000
6.5203644
10.7492072
-----------STRENGTH---------Mean
SD
66.2500000
71.0000000
64.2500000
6
16.6368781
16.9452901
4.8032727
(d) [5] Are there significant differences among the times? Report the appropriate F- and Pvalues to support your answer?
(e) [5] Compute the LSD for comparing means strengths for the three times. What, if any,
times are significantly different?
(f) [5] Is there a significant interaction between temperature and time? Report the appropriate F- and P- values to support your answer.
(g) [5] On the next page is an interaction plot. Interpret this plot and indicate what it tells
you about the combined effects of time and temperature.
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